Xue Xiao

Yiming Yang, Xiaoyuan Cheng, Daniel Giles et al.

Variational data assimilation estimates the dynamical system states by minimizing a cost function that fits the numerical models with the observational data. Although four-dimensional variational assimilation (4D-Var) is widely used, it faces high computational costs in complex nonlinear systems and depends on imperfect state-observation mappings. Deep learning (DL) offers more expressive approximators, while integrating DL models into 4D-Var is challenging due to their nonlinearities and lack of theoretical guarantees in assimilation results. In this paper, we propose Tensor-Var, a novel framework that integrates kernel conditional mean embedding (CME) with 4D-Var to linearize nonlinear dynamics, achieving convex optimization in a learned feature space. Moreover, our method provides a new perspective for solving 4D-Var in a linear way, offering theoretical guarantees of consistent assimilation results between the original and feature spaces. To handle large-scale problems, we propose a method to learn deep features using neural networks within the Tensor-Var framework. Experiments on chaotic systems and global weather prediction with real-time observations show that Tensor-Var outperforms conventional and DL hybrid 4D-Var baselines in accuracy while achieving a 10- to 20-fold speed improvement.

Maida Wang, Xiao Xue, Mingyang Gao et al.

We introduce a quantum-informed machine learning (QIML) framework for the long-term dynamical behavior of high-dimensional chaotic systems. The method combines a one-time, offline-trained quantum generative model with a classical autoregressive predictor for spatiotemporal field generation. The quantum model learns a quantum prior (Q-Prior) that guides the representation of small-scale interactions and improves the modeling of fine-scale dynamics. We evaluate QIML on three representative systems: the Kuramoto-Sivashinsky equation, the two-dimensional Kolmogorov flow, and a cross-section of fully developed three-dimensional turbulent channel flow used as a realistic inflow condition. Compared to the classical baseline, QIML yields up to 17.25% improvement in predictive distribution accuracy and a 29.36% improvement in the fidelity of the predicted full energy spectrum. For turbulent channel inflow, the Q-Prior is essential: without it, the model fails to evolve in time, while QIML produces stable, physically consistent forecasts that surpass leading machine learning models for PDEs, including the Fourier Neural Operator and Markov Neural Operator, whose errors diverge. Beyond accuracy, QIML also achieves a memory advantage, compressing multi-megabyte datasets into a kilobyte-scale Q-Prior that captures only the invariant measure needed to guide the classical model, thus circumventing Holevo's bound by avoiding full data reconstruction. Our findings provide a practical and scalable pathway for integrating the advantages brought by quantum devices into large-scale scientific, engineering modeling and simulation.